#1572449

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1660 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(2,4)(6,8)(11,15)(14,16)(19,23)(20,24)(25,26,30,31)(27,33,35,28)(29,37,38,32)(34,36,40,39);(3,7)(6,8)(9,10,13,12)(11,14,15,16)(17,18,21,22)(19,24,23,20)(26,31)(28,33)(29,38)(34,40);(1,2,5,4)(3,6,7,8)(9,10,13,12)(11,14,15,16)(17,19)(18,20)(21,23)(22,24)(25,27,30,35)(26,28,31,33)(29,34,38,40)(32,36,37,39);(1,3,5,7)(2,6,4,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,20)(21,22)(23,24)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,40)(38,39);(1,4,5,2)(3,8,7,6)(9,12,13,10)(11,16,15,14)(17,20)(18,19)(21,24)(22,23)(25,29,30,38)(26,32,31,37)(27,34,35,40)(28,36,33,39)

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=ghi,d^2=fgh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1479645);